My goal is to compute

\begin{equation} I = \det \left(\mathbf{I} + \mathbf{A}\right) \end{equation} where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers: $$ \left\{ \begin{array}{crc} \mathbf{A}_{ij} = C_{p-1} \alpha^{2(p-1)} &\mbox{ if }& i+j=2p \mbox{, and with }C_p= \frac{1}{p+1}\binom {2p} {p}\\ \mathbf{A}_{ij} = 0 &\mbox{ if }& i+j \mbox{ is odd.} \end{array} \right. $$ with $\alpha >0$ a parameter.

Numerically, it seems that $I$ has a limit when $n\to +\infty$ if $\alpha<1/2$ and diverges to $\infty$ otherwise.

Any idea ?